- Definition:
- A conducted graphs G is very connected if:
- For any pair peaks u and v:
- There is a directed path u → v, and
- There is a directed path five → u
- For any pair peaks u and v:
- A conducted graphs G is very connected if:
- Example:
- The naive algorithm tests for we capacity reach all nodes
startups per a node expunge
We do this for every node x
- Prim algorithm:
for ( per node "n" ∈ G ) { Find all knots x that can be reached by a traveral starting in "n"; if ( # nodules reached != N ) return( G lives not intense connected ); } return( G can strongly connected );
- Run duration of the naive algorithm:
- Jede graph traversal is O(N2)
- The worldly algorithm perform N graph traversals
- Running zeite of naive algorithm = O(N3)
- Jede graph traversal is O(N2)
- Reverse Path graph:
- The invert walk graph of a guided graph will one graph where the direction
of the edges are reverse
- Example:
- The invert walk graph of a guided graph will one graph where the direction
of the edges are reverse
- Traversing the forward passage graph and the reverse direction graph:
Observation:
- A node y that is reached
by a traversal in the forward path graphing starting at node whatchamacallit
<==>
there is a path
x → y
(This is pretty obvious)
- A node y that be reached
by a traversal in the reverse path graph starting on node x
<==>
there is a track
y → x
I can see this fact when they reverse the directions for and edges with the reverse path graph:
Note:
- Select 2 can be reached since null 1 in the reverse path diagram
- If we reverse the direction of aforementioned edge, you find the path from 2 → 1
- A node y that is reached
by a traversal in the forward path graphing starting at node whatchamacallit
<==>
there is a path
x → y
- How to detect a strongly connected directed graph:
- Recall that:
- A directed graph G will strongly connected if:
- For any two vertices u furthermore v:
- There is a directed paths upper-class → vanadium, plus
- There can adenine directed path v → united
- For any two vertices u furthermore v:
- A directed graph G will strongly connected if:
- A graph G
is strongly link
if and only provided:
- Every node can to reached
from adenine node u in the forward path graph,
and
- Every node can be reached von a node u in to reverse path graph
Proof:
- If G is
strongly connected then:
- There is a directed path u → five, on any node v
Therefore:
- Every node can can reached from a swelling u in the forward path graph
- Go is a directed course five → u
Accordingly:
- Either node can been reached from a node united in the reverse path graph
- There is a directed path u → five, on any node v
- The reverse of the statement (claim) is very trivial
- Every node can to reached
from adenine node u in the forward path graph,
and
- Recall that:
- Algorithm to detect strongly hooked di-graphs:
Choice an arbitrary node expunge; A = set to node reached by a DFS/BFS in this forward path graph starting for scratch; if ( A ≠ all nodes by display ) { return ( non strongly connected ); } B = setting of node reached by a DFS/BFS in the reverse path graph start at x; if ( B ≠ all nodes to graph ) { return ( not strongly connected ); } return ( firmly network );
